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Expressing uncertainty
There are two ways that the uncertainty in a quantity can be expressed – as an absolute value and as a percentage.
Absolute uncertainty
Absolute uncertainty refers to the raw margin of error in any measurement (i.e. the "give or take").
Example:
A car is measured to be 2.8 m long, give or take 0.1m. We can write the length of the car as:
L = 2.8 m ± 0.1 m
Where 0.1m is said to be the absolute uncertainty.
A car is measured to be 2.8 m long, give or take 0.1m. We can write the length of the car as:
L = 2.8 m ± 0.1 m
Where 0.1m is said to be the absolute uncertainty.
Example:
John is between 174.5 cm and 175.5 cm tall. Another way of thinking of this, is that John's height is within 0.5cm of 175cm. We can therefore write John's height as:
Height = 175 cm ± 0.5 cm
Where 0.5cm is the absolute uncertainty.
John is between 174.5 cm and 175.5 cm tall. Another way of thinking of this, is that John's height is within 0.5cm of 175cm. We can therefore write John's height as:
Height = 175 cm ± 0.5 cm
Where 0.5cm is the absolute uncertainty.
Percentage uncertainty
It is often useful to write the uncertainty as a percentage of the measurement. If we measure something to be x, with an absolute uncertainty of Δx, then:
Percentage uncertainty = (Δx / x) × 100%
Example:
In the example above the car was measure to be 2.8m long with an absolute uncertainty of 0.1m. Since 0.1m is 4% of 2.8 m, the percentage uncertainty in the cars length is 4%. We therefore can write the length of the car with the percentage uncertainty as:
L = 2.8 m ± 4%
In the example above the car was measure to be 2.8m long with an absolute uncertainty of 0.1m. Since 0.1m is 4% of 2.8 m, the percentage uncertainty in the cars length is 4%. We therefore can write the length of the car with the percentage uncertainty as:
L = 2.8 m ± 4%
Rounding uncertainties
- Round both absolute and percentage uncertainties to 1 s.f.
- Then round the measurement to the same number of decimal places as the absolute uncertainty
Example:
2.5764 ± 0.8453 (unrounded)
Written as absolute uncertainty:
2.6 ± 0.8
Written as percentage uncertainty:
(0.8453/2.5764) × 100% = 32.81% = 30% (1sf)
2.6 ± 30%
2.5764 ± 0.8453 (unrounded)
Written as absolute uncertainty:
2.6 ± 0.8
Written as percentage uncertainty:
(0.8453/2.5764) × 100% = 32.81% = 30% (1sf)
2.6 ± 30%
Example:
1234 ± 456 (unrounded)
Written as absolute uncertainty:
1200 ± 500
Written as percentage uncertainty:
(456/1234) × 100% = 36.95% = 40% (1sf)
1200 ± 40%
1234 ± 456 (unrounded)
Written as absolute uncertainty:
1200 ± 500
Written as percentage uncertainty:
(456/1234) × 100% = 36.95% = 40% (1sf)
1200 ± 40%